1

Unified Growth Theory Contradicted by the

GDP/cap Data Ron W Nielsen

1

Environmental Futures Research Institute, Gold Coast Campus, Griffith University, Qld,

4222, Australia

September, 2015

Mathematical properties of the historical GDP/cap distributions are discussed and

explained. These distributions are frequently incorrectly interpreted and the

Unified Growth Theory is an outstanding example of such common

misconceptions. It is shown here that the fundamental postulates of this theory are

contradicted by the data used in its formulation. The postulated three regimes of

growth did not exist and there was no takeoff at any time. It is demonstrated that

features interpreted as three regimes of growth represent just mathematical

properties of a single, monotonically-increasing distribution, indicating that a

single mechanism should be used to explain the historical economic growth. It is

shown that using different socio-economic conditions for different perceived parts

of the historical GDP/cap data is irrelevant and scientifically unjustified. The

GDP/cap growth was indeed increasing slowly over a long time and fast over a

short time but these features represent a single, uniform and uninterrupted growth

process, which should be interpreted as whole using a single mechanism of

growth.

Introduction

Hyperbolic distributions appear to be creating significant problem with their interpretation.

They are routinely seen as being made of two distinctly different components, slow and fast,

jointed by a transition stage. However these distributions are easy to understand if they are

represented by their reciprocal values (Nielsen, 2014) because in this representation the

confusing features disappear and hyperbolic distributions are represented by straight lines.

Significantly more difficult problem is to understand the distributions representing the

historical Gross Domestic Product per capita (GDP/cap) because features, which were

already difficult to understand for hyperbolic distributions, are even more confusing. In

addition, these distributions cannot be simplified by their reciprocal values and to understand

them we have to use a different approach.

Incorrect interpretations of the historical GDP/cap data is a serious problem and the

prominent example is the Unified Growth Theory (Galor, 2005a; 2011). Using the reciprocal

values of the GDP data, it has been already demonstrated (Nielsen, 2014) that the

fundamental postulates of this theory are contradicted by empirical evidence. We shall now

demonstrate that the same conclusion can be reached by analysing the GDP/cap data coming

from precisely the same source as used in developing this theory.

1AKA Jan Nurzynski, r.nielsen@griffith.edu.au; ronwnielsen@gmail.com;

http://home.iprimus.com.au/nielsens/ronnielsen.html

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The Unified Growth Theory is aimed at explaining the apparent different stages of growth but

we shall demonstrate that this explanation is irrelevant because the GDP/cap data follow a

single trajectory, indicating that they should be explained using a single mechanism. We shall

demonstrate that the three regimes of growth did not exist and that there was no takeoff in the

economic growth at any time.

Crude representation of data

The GDP/cap distributions are frequently displayed in a grossly simplified way by selecting

just four strategically-located points (Ashref, 2009; Galor, 2005a, 2005b, 2007, 2008a,

2008b, 2008c, 2010, 2011, 2012a, 2012b, 2012c; Galor and Moav, 2002; Snowdon & Galor,

2008) as shown in the top panel of Figure 1. In this figure we show an example for the world

economic growth but similar plots are also used for the regional data. Such displays are

strongly suggestive and they serve as a perfect prescription for drawing incorrect conclusions.

However, plotting more data in a certain way can be also strongly suggestive and confusing

(see the lower panel of Figure 1).

Figure 1. Gross Domestic Product (GDP) per capita (Maddison, 2001) as frequently plotted

(Ashref, 2009; Galor, 2005a, 2005b, 2007, 2008a, 2008b, 2008c, 2010, 2011, 2012a, 2012b,

2012c; Galor and Moav, 2002; Snowdon & Galor, 2008) to explain the mechanism of the

historical economic growth.

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The GDP/cap distributions are already sufficiently confusing even if all data are plotted

without joining them by straight lines. They suggest a prolonged epoch of stagnation

represented by the low values of the GDP/cap ratio, followed by a rapid takeoff representing

an alleged new regime of economic growth governed by a distinctly different mechanism.

Such distributions have to be analysed with extra care but plots such as shown in Figure 1 are

not helpful because they reinforce incorrect impressions and interpretations.

Impressions can be misleading and every effort should be taken to avoid being guided by

their deception. Science is not based on impressions but on rigorous analysis of data. Unified

Growth Theory (Galor, 2005a; 2011) describes ideas based on impressions created by such

displays as shown in Figure 1 or by quoting certain data without making any effort to analyse

them scientifically. Incorrect concepts remain incorrect even if translated into mathematical

formulae. It has already been shown that the fundamental concepts of the Unified Growth

Theory are contradicted by the historical GDP data (Nielsen, 2014). We shall now show that

they are also contradicted by the GDP/cap data.

Explaining the GDP/cap ratio

The GDP/cap ratio combines two time-dependent distributions: (1) the time-dependent GDP

growth and (2) the time-dependent population growth. In order to understand the GDP/cap

distributions we have to understand their two components: the growth of the GDP and the

growth of the population.

Over 50 years ago, von Foerster, Mora and Amiot (1960) demonstrated that the world

population was increasing hyperbolically during the AD era. The natural tendency of the

historical GDP growth (not only global but also regional) is also to follow hyperbolic

distributions (Nielsen, 2014, 2015a, 2015b).

Even though a hyperbolic distribution appears to be made of two different components, slow

and fast, joined by a transition component, it has been shown (Nielsen, 2014) that such

interpretation is based on strongly misleading impressions. Reciprocal values of a hyperbolic

distribution describing growth follow a decreasing straight line and it is then obvious that it

makes no sense to divide a straight line into arbitrarily selected sections and claim different

mechanisms of growth for each section. It also makes no sense to look for a point marking a

takeoff on such a monotonically decreasing straight line because a monotonically decreasing

straight line remains a monotonically decreasing straight line and there is no justification in

selecting a certain point on such a line and claim that there is a change of direction at this

point because there is no change of direction.

The first and essential step in trying to understand the GPD/cap data, regional or local is to

check whether their two components (GDP and population) follow hyperbolic distributions.

Based on the available evidence (Nielsen, 2014, 2015a, 2015b; von Foerster, Mora & Amiot,

1960), they are likely to demonstrate such preference. Consequently, in order to understand

the historical GDP/cap data we have to understand the mathematical process of dividing two

hyperbolic distributions.

We are going to demonstrate now that the characteristic features of the GDP/cap

distributions, which were used in the formulation of the Unified Growth Theory (Galor,

2005a, 2011), represent purely mathematical properties of dividing two hyperbolic

distributions. They do not represent different socio-economic conditions describing different

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mechanisms of growth for different perceived sections of these distributions as claimed in the

Unified Growth Theory.

Hyperbolic distribution describing growth is represented by a reciprocal of a linear function:

1( ) ( )f t a kt , (1)

where ( )f t is the size of the growing entity, t is the time, and a and k are positive constants.

A reciprocal of hyperbolic distribution, 1[ ( )]f t , is represented by a decreasing straight line:

1 1[ ( )]

( )f t a kt

f t. (2)

Hyperbolic distributions should not be confused with hyperbolic functions ( sinh( )t , cosh( )t ,

etc). Furthermore, reciprocal distribution or functions, 1[ ( )]f t , should not be confused with

inverse functions, 1( )f t . Mathematical symbol for the inverse function, 1( )f t , is similar to

the mathematical symbol for the reciprocal function, 1[ ( )]f t , but the concepts are different.

In the inverse functions, the roles of variables are inversed. In the reciprocal functions they

remain the same. Thus, for instance, for the distribution given by the equation (1), the

objective of using its inverse function would be to calculate how the time depends on the size

of the growing entity. The inverse of the eqn (1) is

1 1( )

af t

k kt, (3)

where t is now the size of the growing entity and 1( )f t is the time. For the reciprocal

function given by the eqn (2), t is still the time as in the eqn (1). From the eqn (3) we can see

that when the size of the growing entity, t, increases to infinity, the time, 1( )f t , reaches its

terminal value of /a k .

The characteristic feature of hyperbolic distributions is that they increase slowly over a long

time and fast over a short time, escaping to infinity at a certain fixed time /st a k , i.e. when

the denominator in the eqn (1) approaches its zero value. However, as we have already

pointed out and as discussed earlier (Nielsen, 2014), it is a mistake to interpret such

distributions as being made of two distinctly different components joined by a transition

component. It is one and continuous distribution, which has to be interpreted as a whole. If

such a distribution represents a certain mechanism of growth, it is the same mechanism for

the whole distribution.

Let us now take two hyperbolic distributions, ( )f t and ( )g t , and let us divide them. Results

are presented in Figure 2.

Parameters describing hyperbolic distributions displayed in Figure 2 are: 4.5a and 32.2 10k for ( )f t and 7a and 33.35 10k for ( )g t . These distributions are purely

mathematical entities. They have nothing to do with the growth of the population or with the

economic growth. However, they satisfy a simple condition: the singularity of the ( )f t

distribution occurs earlier than the singularity of the ( )g t distribution. For the curves

displayed in Figure 2 singularities are at 2045st for ( )f t and 2090st for ( )g t . The point

of singularity for the ( ) / ( )f t g t ratio is, of course, at 2045st .

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When the distribution ( )f t is divided by ( )g t they produce a distribution, which resembles

closely a typical GDP/cap distribution (see the lower panel of Figure 1). The characteristic

features of this distribution are a long stage of nearly constant values of the ( ) / ( )f t g t ratio

followed by a nearly vertical increase.

Figure 2. Two, mathematically-defined, hyperbolic distributions, ( )f t and ( )g t , and their

ratio ( ) / ( )f t g t . The time of the perceived, but non-existent, takeoff is indicated.

It is important to understand that these features are more pronounced than for the

corresponding hyperbolic distributions. The nearly horizontal part is flatter and the nearly

vertical part is even more vertical. That is why, if the hyperbolic distributions are already so

confusing, the distributions representing the ratio of two hyperbolic distributions are even

more confusing and their interpretation is even more difficult. They have to be analysed with

extra care and their analysis cannot be simplified by using their reciprocal values because the

reciprocal of the ratio of two hyperbolic distributions is also a ratio of two hyperbolic

distributions. Their analysis is significantly more difficult than the analysis of hyperbolic

distributions. They represent a well-concealed trap suggesting strongly the existence of two

or even three different components and even the most experienced researcher, who is not

familiar with hyperbolic distributions or who is reluctant to accept them because of their

singularity, can be easily misguided.

So we can see now that by dividing two, mathematically defined distributions, which have

nothing to do with the economic growth, we have generated the fundamental features, which

inspired the creation of the Unified Growth Theory (Galor, 2005a, 2011): “the Malthusian

Regime” represented by the flat “part,” “Sustained-Growth Regime” represented by the steep

“part,” and the middle “part” which we could call “the Post-Malthusian Regime,” containing

also a perceived “takeoff,” i.e the apparent fast transition from the flat to the steep growth.

These features were generated using purely mathematical, monotonically-increasing

distributions and a mathematical division of these distributions. They reflect purely

mathematical properties of a single distribution representing the ( ) / ( )f t g t ratio. They do not

describe different stages of growth. Furthermore, it is clear that these features cannot be

6

attributed uniquely to the GDP/cap distributions. The division of two hyperbolic distributions

may represent a certain mechanism of growth but it is still a single mechanism.

We have created an unusual and perhaps puzzling distribution but it would be incorrect to be

so mesmerised by this simple mathematical operation as to propose different regimes of

growth for different perceived parts of the ( ) / ( )f t g t ratio. We can see that the features

observed for the GDP/cap distributions can be easily replicated by dividing two

mathematically-defined hyperbolic distributions. It is, therefore, clear that hasty assumptions

about different socio-economic conditions for the different perceived “parts” of the GDP/cap

distributions can be questioned, which means that the whole Unified Growth Theory based on

such assumptions can be also questioned.

The next step in explaining the GDP/cap distributions is now to explain why the division of

two hyperbolic distributions generates such a puzzling trajectory, which appears to be made

of two distinctly different components and why these apparently different components are so

strongly pronounced.

Explaining the ratio of hyperbolic distributions

Using the eqns (1) and (2) we can see that the ratio of two hyperbolic distributions can be

represented also in two other ways:

11

1

( )[ ] [ ( )] [ ][ ( )] [ ] ( )[ ]

( )[ ] [ ( )] [ ]

f t Hyperbolic g t Linearg t Linear f t Hyperbolic

g t Hyperbolic f t Linear (3).

These operations are represented graphically in Figure 3. We can see that all these

mathematical operations create the same distribution representing the ratio ( ) / ( )f t g t . It does

not matter which pathway we take – results are the same.

Dividing two monotonically-increasing hyperbolic distributions is the same as multiplying

hyperbolic distribution by a decreasing linear function and the same as dividing two

decreasing linear functions. It is all just as simple as that. There are no hidden mysteries that

need to be explained by some kind of complicated theories and mechanisms, but we still want

to understand why these simple operations generate such a peculiar distribution, which

appears to be made of two distinctly different components: horizontal and vertical.

The easiest way to understand the division of hyperbolic distributions is probably by looking

at the middle section of Figure 3. The effect of the multiplication of hyperbolic distribution

by the decreasing linear function is to lift up the left-hand part of the slowly increasing

section of hyperbolic distribution and suppress the right-hand part. However, if ( )f t escapes

to infinity earlier than ( )g t , ( )f t will be escaping to infinity when 1[ ( )]g t is still positive.

The values of 1[ ( )]g t will be small but the multiplication of the rapidly increasing values of

( )f t by small values of 1[ ( )]g t will have no effect on the escape to infinity. The product of

such numbers will be also rapidly escaping to infinity. The combined effect of such

multiplication of a decreasing straight line by the increasing hyperbolic distribution is to

flatten the slowly increasing section of the hyperbolic distribution without significantly

changing the large values. The initial slow increase is made even slower and the perceived

transition to the steep part is even more pronounced. However, there is no mathematically-

defined transition at any time.

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Figure 3. Graphic representation of the eqns (3).

The ratio of two hyperbolic distributions can be described simply as the linearly-modulated

hyperbolic distribution. Thus in our example the ratio of ( ) / ( )f t g t can be described as the

linearly-modulated hyperbolic ( )f t distribution. The linear modulation is done by the linear

function 1[ ( )]g t representing the reciprocal values of the hyperbolic ( )g t distribution.

Likewise, the distribution representing the historical GDP/cap growth can be described as the

linearly-modulated hyperbolic GDP distribution. The linear modulation is done by the linear

distribution representing the reciprocal values of the hyperbolic distribution describing the

growth of human population.

The ratio of two hyperbolic distributions looks as if being made of two different components,

slow and fast, but it is still the same, uninterrupted, monotonically increasing distribution. It

is still a single mathematical distribution. It is the distribution, which is not made of two

different sections. It is the distribution that it is impossible to divide into two distinctly

different parts represented by two different functions. This distribution increases slowly over

a long time and fast over a short time but the transition from the perceived slow to the

perceived fast growth occurs over the entire range of time. It is impossible to determine the

time of this perceived transition. It is impossible to determine the time of the perceived

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takeoff because the takeoff does not exist even if it appears to exist. The perceived takeoff is

an illusion. There is a slow growth over a long time and a fast growth over a short time but

there is no transition at any time between the slow and the fast growth. The slow and the fast

growth are represented by the same, monotonically increasing distribution, which is not made

of distinctly different components.

Even though the ratio of hyperbolic distributions, ( ) / ( )f t g t , looks as if it were made of two

or three components (see Figures 2 and 3), even though the distribution represented by this

ratio increases slowly over a long time and fast over a short time, even though it increases to

infinity at a fixed time and even though it appears to be characterised by a takeoff at a certain

time, it is still just a single, monotonically-increasing distribution, which is impossible to

divide into different components. We have to accept it and learn to live with it.

Figure 4. The gradient and growth rate of the ratio of hyperbolic distributions ( ) / ( )f t g t . The

onset of the perceived takeoff shown in Figure 2 is indicated. This figure shows that the

takeoff never happened and that the distribution representing the ratio ( ) / ( )f t g t is not made

of different components. It is a single, monotonically-increasing distribution.

The easiest way to understand this apparent paradox and to dispel the strong illusion of

different components is to examine the reciprocal values of each of the two components of

the ratio ( ) / ( )f t g t , i.e. the reciprocal values 1[ ( )]f t and

1[ ( )]g t as shown in the lowest part

of Figure 3. It would be obviously unreasonable to claim that each of these straight lines is

made of two or three distinctly different components and it would be unreasonable to claim

different mechanisms of growth for various, arbitrarily-selected parts of a straight line. At

which point located on the straight line one mechanism of growth is supposed to end and a

new mechanism to begin? It is impossible to claim two or three distinctly different sections

on the monotonically decreasing straight lines. There is also obviously no feature on such

straight lines that could be claimed as marking a takeoff.

We can also take a different approach and demonstrate that the ratio ( ) / ( )f t g t represents a

single, monotonically-increasing distribution and that there is no takeoff at any time. This

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different approach consists in calculating the gradient and the growth rate of the ( ) / ( )f t g t

ratio. Results are presented in Figure 4 around the time of the perceived takeoff, i.e. when the

( ) / ( )f t g t reaches the value of 2 (see Figure 2). For better clarity, results are plotted as a

function of the size of the ( ) / ( )f t g t ratio.

These calculations show clearly that both the gradient and the growth rate of the hyperbolic

ratio ( ) / ( )f t g t increase monotonically. The perceived takeoff never happened. What looks

like a takeoff in Figure 2 is in fact just the continuation of the undisturbed and monotonically-

increasing distribution representing the ( ) / ( )f t g t ratio. It is impossible to claim different

components for any of the distributions displayed in Figure 4, representing the ( ) / ( )f t g t

distribution, which in Figure 2 looks very deceptively as being made of two different

components. It is impossible to claim a takeoff for any these two distributions. The two

components simply do not exist and the takeoff is just an illusion.

Analysis of the historical GDP/cap data

The GDP and population data (Maddison, 2001) [the same data as used but not analysed

during the formulation of the Unified Growth Theory (Galor, 2005a, 2011)] together with

their fitted hyperbolic distributions are shown in Figure 5. Indicated in the figure is the time

of the Industrial Revolution 1760-1840 (Floud & McCloskey, 1994), which is generally

claimed as the time of the alleged takeoff in the economic growth (Galor, 2005a, 2008a,

2011, 2012). Parameters fitting the GDP data are: 21.716 10a and 68.671 10k while

parameters fitting the population data are 8.724a and 34.267 10k .

Points of singularity are: 1979st for the world GDP and 2045st for the population data.

The point of singularity for the world GDP is before the point of singularity for the growth of

the world population. Consequently, the GDP/cap ratio should display the same features as

shown in Figure 2 for the ( ) / ( )f t g t ratio and indeed it does. The calculated curve and the

data shown in Figure 6 follow a similar distribution as shown in Figure 2.

The point of singularity of the GDP/cap trajectory is determined by the point of singularity

for the GDP distribution. If the point of singularity for the GDP trajectory were higher than

the point of singularity for the population trajectory, the growth of the GDP/cap would have

remained nearly constant over a long time and then it would decrease to zero close to the

point of singularity of the population trajectory. The danger of the escape to infinity of the

GDP/cap growth would have been avoided. Under suitably-chosen conditions the GDP/cap

growth can be made to increase or decrease slowly over a much longer time. If only we

understood the mathematics of the GDP/cap distributions early enough we could have

perhaps been able to control the economic growth in such a way as to ovoid its currently-

experienced rapid and dangerous increase.

The fundamental postulate of the Unified Growth Theory (Galor, 2005a, 2011) is the

existence of three, distinctly different regimes of growth governed by distinctly different

mechanisms: the Malthusian Regime (or Epoch), the Post-Malthusian Regime and the

Sustained-Growth Regime. Galor claims precise timing of these three alleged regimes. The

data (Maddison, 2001) he uses extend only down to AD 1, but he claims that the Malthusian

Regime commenced in 100,000 BC (Galor 2008a, 2012a). He then claims that this regime

ended in AD 1750 for developed countries and in 1900 for less-developed countries. The

Post-Malthusian Regime was between 1750 and 1870 for developed countries and from 1900

for less-developed countries. The Sustained-Growth Regime commenced in 1870 and

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continues until the present time. All these interpretations are not supported by a rigorous and

scientific analysis of data. They were prompted by the incorrect interpretations of hyperbolic

distributions. It appears also that they were fuelled by a good dose of creative imagination,

which was allowed to be uncontrolled by the scientific process of investigation.

Figure 5. Hyperbolic distributions are compared with the world GDP and population data

(Maddison, 2001). The GDP is expressed in billions of 1990 International Geary-Khamis

dollars and the population in billions.

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Figure 6. Calculated, linearly-modulated hyperbolic GDP distribution, representing the

GDP/cap ratio, is compared with the world GDP/cap data (Maddison, 2001). The GDP/cap is

expressed in the 1990 International Geary-Khamis dollars.

If we allow ourselves to be guided by impressions then looking at the data presented in

Figure 1 or even in Figure 6 we would have to conclude that there was a long epoch of

stagnation followed by a sudden explosion, which Galor describes as a takeoff. This

conclusion appears to be obvious and it seems to be supported by the data and by calculated

distribution.

However, the fit to the data is not produced by using different mathematical functions

describing different perceived stages of growth. If we had to use such different functions we

could perhaps claim the existence of different regimes of growth but the fit to the data is

described by a single mathematical distribution.

Furthermore, we already know that there was no takeoff in the growth of the GDP and that

the historical GDP trajectory cannot be divided into three different regimes (Nielsen, 2014).

This conclusion is based on the examination of the reciprocal values of the GDP data.

The same conclusion applies also to the population data, because, as shown in Figure 5, they

are also described by hyperbolic distribution and according to the eqn (2) hyperbolic

distributions describing growth are represented by decreasing straight lines. Dividing a

straight line into different arbitrary sections clearly makes no sense. Likewise, looking for a

change of direction on a straight line to claim a takeoff also makes no sense. Consequently,

we can hardly expect that we can produce a takeoff and three different regimes of growth for

the GDP/cap ratio.

However, we can prove that there was no takeoff for the GDP/cap distribution and that the

three regimes of growth did not exist. We shall do this by calculating the gradient and the

growth rate for the calculated GDP/cap trajectory. These calculations are presented in Figures

7 and 8.

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Figure 7. Gradient of the world GDP/cap calculated using the fitted, linearly-modulated

hyperbolic distribution shown in Figure 6. The GDP/cap is expressed in the 1990

International Geary-Khamis dollars. There was no takeoff at any time and the three regimes

of growth postulated by Galor (2005a, 2011) did not exist.

Figure 8. Growth rate of the GDP/cap calculated using the fitted, linearly-modulated

hyperbolic distribution shown in Figure 6. The GDP/cap is expressed in the 1990

International Geary-Khamis dollars. There was no takeoff at any time and the three regimes

of growth postulated by Galor (2005a, 2011) did not exist.

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A takeoff in the GDP/cap trajectory would be marked by a clear change in the gradient and in

the growth rate around the time of the Industrial Revolution when a transition to a new

economic growth regime was supposed to have happened (Galor, 2005a, 2008a, 2011,

2012a). The shape of the trajectories describing the gradient and growth rate would have to

be distinctly different before and after the Industrial Revolution. There should be a certain

clear discontinuity.

The gradient and the growth rate of the fitted curve increase monotonically confirming that

the fitted, linearly-modulated hyperbolic distribution increases also monotonically. The

calculated curve gives excellent fit to the GDP/cap data and consequently the gradient and the

growth rate of the fitted curve represent also the gradient and the growth rate of the data.

Figures 7 and 8 clearly demonstrate that there is no reason for terminating the alleged

Malthusian Regime around AD 1750 and starting a new regime because there was no unusual

change in the gradient and in the growth rate of the GDP/cap around that time, but there was

also no scientifically-justified reason for assuming the existence of the Malthusian Regime.

There is no reason for terminating the alleged Post-Malthusian Regime around 1870 and

starting the alleged Sustained-Growth Regime. There is no reason for chopping the

monotonically-increasing distributions into three arbitrarily-selected sections. There is no

reason for proposing three regimes of growth governed by distinctly different mechanism.

There is no reason for claiming a takeoff at any time.

These calculations, supported by data, clearly demonstrate that the Industrial Revolution had

no impact on the economic growth trajectory. Impacts were of different kind but the data

show that the Industrial Revolution did not boost the global economic growth. It did not even

boost the economic growth in Western Europe (Nielsen, 2014). The three regimes of growth

did not exist. The fundamental postulates of the Unified Growth Theory (Galor, 2005a, 2011)

are contradicted by the analysis of data, the same data as used but not analysed during the

formulation of this theory. Unified Growth Theory describes and explains phenomena that did

not exist and consequently it does not explain the historical economic growth. It is an

incorrect and misleading theory.

The discussion of socio-economic conditions presented by Galor might be interesting for

another reason but there is no evidence in the GDP data that this discussion has any relevance

for explaining the mechanism of the economic growth. However, even this discussion,

translated repeatedly into mathematical expressions, appears to be dubious and could be also

questioned.

Economic growth was indeed slow over a long time and fast over a short time but it is

incorrect to divide this monotonically increasing distribution into three regimes and claim

distinctly different mechanisms for the arbitrarily selected sections. It is also incorrect to

claim that there was a takeoff at a certain time. The data and their analysis give no scientific

basis for such claims.

Historical economic growth has to be explained using a single mechanism. Such a

mechanism should describe the slow and fast growth including the apparent transition. All

these “parts” should be treated as one. Only then we could claim that we have explained the

mechanism of the historical economic growth.

Dividing the past growth into three different regimes and claiming three different

mechanisms is unsupported by data and it does not explain the mechanism of the historical

economic growth. A truly unified growth theory will have to be based on a single mechanism.

Such an explanation will be proposed in a separate publication (Nielsen, 2015c).

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Summary and conclusions

The aim of our discussion was to explain the puzzling features of the GDP/cap distributions

showing a slow growth over a long time, followed by a rapid increase, the features which

appear to be causing a significant problem with their interpretations, and the outstanding

example is the Unified Growth Theory (2005a, 2011). Our discussion was based on precisely

the same data which were used, but not analysed, during the formulation of the Unified

Growth Theory. These data represent the historical economic growth and the historical

growth of human population. They do not represent the current growth but it is the historical

growth that is causing such a big problem.

Historical economic growth, global and regional, shows a clear preference for increasing

along hyperbolic trajectories (Nielsen, 2014, 2015a, 2015b). Hyperbolic growth contains

singularity, when a growing entity escapes to infinity at a fixed time. However, such a growth

becomes, at a certain stage, impossible and has to be terminated either by catastrophic

collapse or by a diversion to a slower trajectory, which is hardly surprising because many

other types of growth can be and are terminated. For instance, the best known exponential

growth, which does not increase to infinity at a fixed time, becomes impossible after a certain

time and has to be terminated. Historical economic growth was diverted to slower trajectories

between around the end of the 1800s and the mid-1900s (Nielsen, 2015a, 2015b).

Data (Maddison, 2001, 2010) show that the historical hyperbolic growth continued for

hundreds of years (Nielsen, 2015a, 2015b) and we have to accept this evidence because the

hyperbolic growth is uniquely identified by its reciprocal values. It is remarkable that such an

“impossible” type of growth continued for such a long time. However, the mechanism of

growth can change and there is nothing unusual about it. The mechanism of the economic

growth must have changed because while it is no longer following hyperbolic trend it

continues to increase along slower trajectories (Nielsen, 2015a, 2015b). The singularity has

been bypassed and the current economic growth and the growth of human population are

controlled by different mechanism than in the past.

We have discussed mathematical properties of the historical GDP/cap distributions. We have

explained how they should be analysed and interpreted.

If both components of the GDP/cap indicator increase hyperbolically (as for the historical

world economic growth and for the growth of the population) then the GDP/cap distributions

represent a ratio of hyperbolic trajectories. Created features may be easily confusing and

consequently historical GDP/cap data have to be analysed with care. Interpretations based on

impressions and supported by the frequently-used crude display of data (Ashref, 2009; Galor,

2005a, 2005b, 2007, 2008a, 2008b, 2008c, 2010, 2011, 2012a, 2012b, 2012c; Galor and

Moav, 2002; Snowdon & Galor, 2008) represent a perfect prescription for drawing incorrect

conclusions.

We have explained how to understand the confusing features of the historical GDP/cap

distributions. They can be interpreted simply as the linearly-modulated hyperbolic GDP

distributions. Linear modulation is by the reciprocal values of population data. We have

discussed a few ways these distributions can be analysed to understand their characteristic

features.

As an illustration of our discussion, we have investigated the data (Maddison, 2001) used in

developing the Unified Growth Theory (Galor, 2005a, 2011). Earlier investigation (Nielsen,

2014) of the GDP data (Maddison, 2001) revealed that the fundamental postulates of this

theory are unsupported. Now, this conclusion has been confirmed and reinforced by the

analysis of the GDP/cap data.

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In his theory, Galor discusses various socio-economic concepts of growth but his theory does

not explain the mechanism of economic growth because it is based firmly on the

misinterpretation of the purely mathematical features of hyperbolic distributions. His

discussion of socio-economic issues is interesting but it has no relevance for explaining the

mechanism of the economic growth because changes in socio-economic conditions had no

effect on the economic growth trajectory as manifested by the available data (Maddison,

2001), the same data, which were used, but not analysed, in the formulation of the Unified

Growth Theory. Galor’s speculations about socio-economic processes are strongly guided by

phantom features created by hyperbolic illusions.

We have demonstrated that the features interpreted in this theory as different stages of growth

represent in fact just the uniform mathematical properties of the GDP/cap distributions

describing a single-stage economic growth. Unified Growth Theory does not explain the

historical economic growth because the three regimes of growth claimed by this theory did

not exist and there was no takeoff at any time. The theory describes features, which do not

characterise economic growth.

The data (Maddison, 2001) show that the economic growth was indeed slow over a long time

and fast over a short time but the close analysis of these data reveals a single, monotonically-

increasing distribution. There is no need and no justification for breaking the historical

economic growth into different stages of growth. Mathematical analysis of historical

economic growth demonstrates that they have to be explained using a single mechanism.

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